Theorem wdomtr 8480

Description: Transitivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)

Assertion

Ref	Expression
wdomtr	|- ( ( X ~<_* Y /\ Y ~<_* Z ) -> X ~<_* Z )

Proof of Theorem wdomtr

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relwdom 8471	. . . . 5 |- Rel ~<_*
2	1	brrelex2i 5159	. . . 4 |- ( Y ~<_* Z -> Z e. _V )
3	2	adantl 482	. . 3 |- ( ( X ~<_* Y /\ Y ~<_* Z ) -> Z e. _V )
4		0wdom 8475	. . . 4 |- ( Z e. _V -> (/) ~<_* Z )
5		breq1 4656	. . . 4 |- ( X = (/) -> ( X ~<_* Z <-> (/) ~<_* Z ) )
6	4, 5	syl5ibrcom 237	. . 3 |- ( Z e. _V -> ( X = (/) -> X ~<_* Z ) )
7	3, 6	syl 17	. 2 |- ( ( X ~<_* Y /\ Y ~<_* Z ) -> ( X = (/) -> X ~<_* Z ) )
8		simpll 790	. . . . 5 |- ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) -> X ~<_* Y )
9		brwdomn0 8474	. . . . . 6 |- ( X =/= (/) -> ( X ~<_* Y <-> E. z z : Y -onto-> X ) )
10	9	adantl 482	. . . . 5 |- ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) -> ( X ~<_* Y <-> E. z z : Y -onto-> X ) )
11	8, 10	mpbid 222	. . . 4 |- ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) -> E. z z : Y -onto-> X )
12		simpllr 799	. . . . . 6 |- ( ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) /\ z : Y -onto-> X ) -> Y ~<_* Z )
13		simplr 792	. . . . . . . 8 |- ( ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) /\ z : Y -onto-> X ) -> X =/= (/) )
14		dm0rn0 5342	. . . . . . . . . . . 12 |- ( dom z = (/) <-> ran z = (/) )
15	14	necon3bii 2846	. . . . . . . . . . 11 |- ( dom z =/= (/) <-> ran z =/= (/) )
16	15	a1i 11	. . . . . . . . . 10 |- ( z : Y -onto-> X -> ( dom z =/= (/) <-> ran z =/= (/) ) )
17		fof 6115	. . . . . . . . . . . 12 |- ( z : Y -onto-> X -> z : Y --> X )
18		fdm 6051	. . . . . . . . . . . 12 |- ( z : Y --> X -> dom z = Y )
19	17, 18	syl 17	. . . . . . . . . . 11 |- ( z : Y -onto-> X -> dom z = Y )
20	19	neeq1d 2853	. . . . . . . . . 10 |- ( z : Y -onto-> X -> ( dom z =/= (/) <-> Y =/= (/) ) )
21		forn 6118	. . . . . . . . . . 11 |- ( z : Y -onto-> X -> ran z = X )
22	21	neeq1d 2853	. . . . . . . . . 10 |- ( z : Y -onto-> X -> ( ran z =/= (/) <-> X =/= (/) ) )
23	16, 20, 22	3bitr3rd 299	. . . . . . . . 9 |- ( z : Y -onto-> X -> ( X =/= (/) <-> Y =/= (/) ) )
24	23	adantl 482	. . . . . . . 8 |- ( ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) /\ z : Y -onto-> X ) -> ( X =/= (/) <-> Y =/= (/) ) )
25	13, 24	mpbid 222	. . . . . . 7 |- ( ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) /\ z : Y -onto-> X ) -> Y =/= (/) )
26		brwdomn0 8474	. . . . . . 7 |- ( Y =/= (/) -> ( Y ~<_* Z <-> E. y y : Z -onto-> Y ) )
27	25, 26	syl 17	. . . . . 6 |- ( ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) /\ z : Y -onto-> X ) -> ( Y ~<_* Z <-> E. y y : Z -onto-> Y ) )
28	12, 27	mpbid 222	. . . . 5 |- ( ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) /\ z : Y -onto-> X ) -> E. y y : Z -onto-> Y )
29		vex 3203	. . . . . . . . . 10 |- z e. _V
30		vex 3203	. . . . . . . . . 10 |- y e. _V
31	29, 30	coex 7118	. . . . . . . . 9 |- ( z o. y ) e. _V
32		foco 6125	. . . . . . . . 9 |- ( ( z : Y -onto-> X /\ y : Z -onto-> Y ) -> ( z o. y ) : Z -onto-> X )
33		fowdom 8476	. . . . . . . . 9 |- ( ( ( z o. y ) e. _V /\ ( z o. y ) : Z -onto-> X ) -> X ~<_* Z )
34	31, 32, 33	sylancr 695	. . . . . . . 8 |- ( ( z : Y -onto-> X /\ y : Z -onto-> Y ) -> X ~<_* Z )
35	34	adantl 482	. . . . . . 7 |- ( ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) /\ ( z : Y -onto-> X /\ y : Z -onto-> Y ) ) -> X ~<_* Z )
36	35	expr 643	. . . . . 6 |- ( ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) /\ z : Y -onto-> X ) -> ( y : Z -onto-> Y -> X ~<_* Z ) )
37	36	exlimdv 1861	. . . . 5 |- ( ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) /\ z : Y -onto-> X ) -> ( E. y y : Z -onto-> Y -> X ~<_* Z ) )
38	28, 37	mpd 15	. . . 4 |- ( ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) /\ z : Y -onto-> X ) -> X ~<_* Z )
39	11, 38	exlimddv 1863	. . 3 |- ( ( ( X ~<_* Y /\ Y ~<_* Z ) /\ X =/= (/) ) -> X ~<_* Z )
40	39	ex 450	. 2 |- ( ( X ~<_* Y /\ Y ~<_* Z ) -> ( X =/= (/) -> X ~<_* Z ) )
41	7, 40	pm2.61dne 2880	1 |- ( ( X ~<_* Y /\ Y ~<_* Z ) -> X ~<_* Z )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   =/= wne 2794   _Vcvv 3200   (/)c0 3915   class class class wbr 4653   dom cdm 5114   ran crn 5115   o. ccom 5118   -->wf 5884   -onto->wfo 5886   ~<_^* cwdom 8462

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-fun 5890  df-fn 5891  df-f 5892  df-fo 5894  df-wdom 8464

This theorem is referenced by:  wdomen1  8481  wdomen2  8482  wdom2d  8485  wdomima2g  8491  unxpwdom2  8493  unxpwdom  8494  harwdom  8495  pwcdadom  9038  hsmexlem1  9248  hsmexlem4  9251